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2 Ju n 20 07 On the Linearization of the Painlevé Equations
| Content Provider | Semantic Scholar |
|---|---|
| Author | Joshi, Nalini T. Kitaev, Alexander V. Treharne, P. A. |
| Copyright Year | 2007 |
| Abstract | We extend similarity reductions of the coupled (2+1)-dimensional three-wave resonant interaction system, to its Lax pair. Thus we obtain new 3×3 matrix Fuchs–Garnier pairs for the third and fifth Painlevé equations, together with the previously known Fuchs– Garnier pair for the fourth and sixth Painlevé equations. These Fuchs–Garnier pairs have an important feature: they are linear with respect to the spectral parameter. Therefore we can apply the Laplace transform to study these pairs. In this way we found reductions of all pairs to the standard 2×2 matrix Fuchs–Garnier pairs obtained by M. Jimbo and T. Miwa. As the application of the 3×3 matrix pairs, we found integral auto-transformation for the standard Fuchs–Garnier pair for the fifth Painlevé equation. It generates Okamotolike Bäcklund transformation for the fifth Painlevé equation. Another application is an integral transformation relating two different 2 × 2 matrix Fuchs–Garnier pairs for the third Painlevé equation. 2000 Mathematics Subject Classification: 34M55, 33E17, 33E30. PACS 2006: 02.30.Ik, 02.30.Gp, 02.30. Hq. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://arxiv.org/pdf/0706.1750v1.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
